10648 19683 32768 50653 74088
12167 21952 35937 54872 79507
13824 24389 39304 59319 85184
15625 27000 42875 64000 91125
17576 29791 46656 68921 97336
The numbers at third column and third row must have the same middle digit, which will go in the centre square of the grid. So, the choices for these to numbers come down to
The first digits of third column and third row are the middle digits of first row and first column respectively. Since neither 2 nor 4 is the middle digit of any number in the above table, we can eliminate 27000, 24389, 46656, 29791, 42875, and 21952 for the next level. This leaves only one choice if the central digit is 7. So the second table reduces to the following table.
If the central digit is taken as 0, then third column number is 64000 and third row number is 74088 OR third column is 74088 and third row number is 64000. In both cases first row and first column must have the same first digit and their middle digits must be 6 and 7. There is no such pair of numbers in the first table. So the central digit cannot be 0.
By a similar argument, the central digit cannot be 1 or 5.
Now we are left with the possibilities for 3, 6, 8, 9, together with their reflections about the line through the grid from top left to bottom right.
The last columns of first three tables above cannot be filled from the data we have hence the answer for this puzzle is
and
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